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nt. J. Services and Operations Management, Vol. 21, No. 2, 2015 217
Copyright © 2015 Inderscience Enterprises Ltd.
A unique support vector regression for improved
modelling and forecasting of short-term gasoline
consumption in railway systems
Ali Azadeh*, Azam Boskabadi and
Shima Pashapour
School of Industrial Engineering,
Center of Excellence for Intelligent Based Experimental Mechanic,
College of Engineering,
University of Tehran,
P.O. Box 11155-4563, Tehran, Iran
Email: aazadeh@ut.ac.ir
Email: Azam.boskabadi@gmail.com
Email: shima.pashapour@ut.ac.ir
*Corresponding author
Abstract: This study presents a support vector regression algorithm and
time series framework to estimate and predict weekly gasoline consumption in
railway transportation industry. For training support vector machines, recursive
finite Newton (RFN) algorithm is used. Furthermore, it considers the effect
of number of holidays per weeks and amount of transported freight and number
of transported passengers in gasoline consumption prediction. Transported
passengers per kilometre and transported tons per kilometre are the most
important factors in railway industry. For this reason, this study assesses the
effect of these factors on weekly gasoline consumption. Weekly gasoline
consumption in railway transportation industry of Iran from August 2009 to
December 2011 is considered. It is shown that SVR achieves better results in
comparison with other intelligent tools such as artificial neural network (ANN).
Keywords: support vector regression; SVR; gasoline consumption;
forecasting; railway system; artificial neural network; ANN.
Reference to this paper should be made as follows: Azadeh, A., Boskabadi, A.
and Pashapour, S. (2015) ‘A unique support vector regression for
improved modelling and forecasting of short-term gasoline consumption in
railway systems’, Int. J. Services and Operations Management, Vol. 21, No. 2,
pp.217–237.
Biographical notes: Ali Azadeh is an Eminent University Professor and
Founder of Department of Industrial Engineering and Co-founder of Research
Institute of Energy Management and Planning at University of Tehran. He
obtained his PhD in Industrial and Systems Engineering from the University of
Southern California. He received the 1992 Phi Beta Kappa Alumni Award
for Excellence in Research and Innovation of Doctoral Dissertation in USA. He
is the recipient of six awards at the University of Tehran. He is also the
recipient of National Eminent Researcher Award in Iran. He has published
more than 650 papers in reputable academic journals and conference
proceedings.
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Azam Boskabadi received her MSc from the School of Industrial Engineering
at University of Tehran. Her current research includes logistics planning,
complex adaptive systems, agent-based modelling and simulation.
Shima Pashapour is a PhD student at the School of Industrial Engineering at
University of Tehran. Her current research includes operations research,
complex adaptive systems, agent-based modelling and simulation.
1 Motivation and significance
This study presents an intelligent algorithm to model time series data with respect to
gasoline consumption in railway transportation industry in Iran. This is the first study that
integrates conventional time series and support vector regression (SVR) for forecasting
and modelling gasoline consumption in railway industry in Iran. The superiority of the
proposed algorithm is shown by comparing its results with other intelligent tools such as
artificial neural network (ANN).
2 Introduction
Transportation is the main factor for evaluating the industrial improvement of a country.
Gasoline, as one of the most important resources of energy, with its ever growing role in
world economy and transportation has gained special attention more than before.
Through the development of societies and growth of transportation activities, optimum
use of fuel has considered more effective factor in improvement of corporations and their
services. Because of this reason, a major topic applicable in this area is the estimation of
the gasoline consumption, which reveals the consumption growth in the forthcoming
years. Su (2011) considered the effect of population density, freeway road density, and
congestion on household gasoline consumption by using semiparametric and parametric
approaches. Results showed that areas with higher freeway densities, higher levels of
congestion, or lower population densities consume more gasoline. Tasdemir et al. (2011)
used ANN and fuzzy expert system (FES) modelling of a gasoline engine to predict
engine power, torque, specific fuel consumption and hydrocarbon emission. Results show
that developed ANN and FES can be used reliably in automotive industry and
engineering instead of experimental work. Coyle et al. (2012) estimate supply and
demand functions for gasoline using information from excise tax. Results show that
raising fuel taxes will generate significant amounts of revenue with relatively low
efficiency costs. Crôtte et al. (2010) estimated cross elasticities of the demand for
gasoline per vehicle using both a time series cointegration model and a panel GMM
model for Mexican states. Results show that more fuel efficient technologies have a
negligible effect on gasoline consumptions and vehicle stock size has a higher impact
on gasoline consumption. Wadud et al. (2010) model US gasoline demand using
semiparametric techniques. Results show that households located in urban areas reduce
consumption more than those in rural areas in response to an increase in price.
Pock (2010) used various dynamic panel estimators to estimate gasoline demand. Results
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show that standard pooled estimators are more reliable than common IV/GMM
estimators.
This study presents an intelligent integrated algorithm to model time series data with
respect to gasoline consumption in railway transportation industry in Iran. The main
contribution in this study is that this is the first study integrates conventional time series
and SVR for forecasting and modelling gasoline consumption in railway industry in Iran.
Our purpose is forecasting the amount of gasoline that will be used for CEOs’ long-term
decision-making strategies in Iran railway transportation industry. We are supposed to
answer these questions:
• How can heavy industries like a railway industry predict their main resources of
energy and which factors are more important in this procedure?
• If these industries want to predict their usage of energy in the future, which approach
is the best approach that can recognise patterns and analyse data?
• How close is the result of the selected approach to the real amount?
Main factors that are considered in this paper for prediction of gasoline consumption are
number of holidays in weeks, amount of transported loads and, and number of
transported passengers. Transported passengers per kilometre and transported tons per
kilometre are the most important factors in the railway industry. This study considers the
weekly data from August 2009 to December 2011 to show the applicability and
superiority of the proposed algorithm.
3 Literature review
There are a lot of papers in literature that worked on railway industry. Azadeh et al.
(2012) presented an integrated fuzzy modelling and simulation approach for modelling
and scheduling of cargo and passenger trains with time limitations. Rayeni and Saljooghi
(2014) developed a new secondary goal based on symmetric weight selection of
cross-efficiency for ranking and measuring efficiency of railway in Iran. Khare and
Handa (2011) conducted an exploratory research to study customer experience of the
online reservation system of the Indian railways. In this paper, SVR is used for improved
modelling and forecasting of short-term gasoline consumption in railway systems.
Some papers considered prediction of gasoline consumption. Togun and Baysec
(2010b) presented an ANN model to predict the torque and brake gasoline consumption.
They developed a model based on back propagation algorithm. They showed that their
model had high efficiency and accuracy. Togun and Baysec (2010a) presented a model
based on genetic programming (GP) for generating the formulations for gasoline engine
torque and brake specific gasoline consumption. The proposed model show very good
agreement with the experimental results. The results of proposed model compared with
neural network model and strongly good agreement was observed between two
predictions. Azadeh et al. (2010) presented an adaptive intelligent algorithm for
forecasting gasoline demand based on ANN, regression, and design of experiment
(DOE). The results show that ANN provides far less error than regression. Park and Zhao
(2010) estimated US gasoline demand using a time-varying regression.
A new technique in time series forecasting is support vector machines (SVMs)
(Mukherjee et al., 1997; Muller et al., 1999, 1997) and because time series prediction is
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like an auto-regression in time, a regression method can be applied for this task (Thissen
et al., 2003). SVMs illustrate a recent AI model developed by Vapnik (1995) that has
already been used in a vast range of applications. SVMs, as a set of related supervised
learning methods, are a learning tool for solving classification and regression problems
that analyse data and recognise patterns. A SVM makes a hyper plane or set of hyper
planes in a high or infinite dimensional space, which can be used for classification,
regression or other tasks. SVM implements by matching input vectors into a higher
dimensional feature space. The optimum hyper plane is identified through this feature
space with the help of a kernel function, K(x). This inner product in the feature space
makes training data linear-separable. The selected kernel function should satisfy Mercer’s
condition that determines if a prospective kernel is actually an inner product in
some space and guarantees that unique global optimum solutions are obtained (Burges,
1998). Different kinds of kernel functions used today such as linear kernel, the
polynomial kernel, radial basis function (RBF) [include Gaussian radial basis function
(GAUSSIANRBF), exponential RBF and MLP], and sigmoid kernel. But the RBF kernel
has been proposed by the most of users as the best choice, because of its ability to analyse
higher dimension data, use of just one hyper parameter to search, and fewer numerical
difficulties (Hsu et al., 2003). As a large number of researchers such as Burges (1998),
Hao (2003) and Wang et al. (2005) stated that SVMs had a higher performance than
traditional learning tools, SVMs capability and prediction accuracy are determined by the
optimal penalty and kernel parameters.
A novel version of SVM was proposed by Vapnik (1995), Burges (1998) and Smola
and Schoolkopf (2004). It uses a regression method for modelling and prediction. This
method is called SVR.
The model made by SVMs classification depends only on a subset of the training
data, because the cost function for making the model does not consider training points
that lie beyond the margins. Similarly, the model made by SVR depends only on a subset
of the training data and its cost function for building the model ignores any training data
close to the model prediction (within a threshold ε).
As mentioned SVMs and kernel methods (KMs) have become one of the most
popular approaches for learning from examples in science and engineering. They include
some methods for generalisation improvement, model selection, hyper-parameter tuning
as well as a priori knowledge. Sanchez (2003) presented the basic methodology of SVMs
and KMs, and also their utilisation that are classification, regression, clustering, density
estimation, and novelty detection. At the end, he discussed the area of application of
SVMs and KMs.
Tay and Cao (2002) have presented a modified version of SVMs for modelling
non-stationary financial time series that called C-ascending SVM. They obtained this
method by a simple modification of the regularised risk function in support vector
machines. “The empirical risk function has equal weight C to all the ε_insensitive errors
between the predicted and actual values” in the standard SVMs. The regularisation
constant C plays the trade-off role between the empirical risk and the regularised term. So
they applied a variable amount for C. They concluded that the C-ascending support
vector machine has better forecasting results for the actually ordered sample data than the
standard SVMs.
Cao (2003) have proposed using the SVMs experts for time series anticipating. There
are two stages neural network architecture in this kind of support vector machines. In
the first stage, the whole input space is divided into several disjointed regions by
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self-organising feature map (SOM) as a clustering algorithm. Then, in the second step,
SVM constructs the best fit partitioned regions by finding the most appropriate kernel
function and the optimal free parameters of SVMs.
Thissen et al. (2003) predicted time series for the field of chemo metrics by
performing SVMs, Elman recurrent neural networks, and autoregressive moving average
(ARMA) models. The authors have applied ε-SVR method for forecasting future data.
They have considered three datasets; ARMA time series, the chaotic and non-linear
Mackey-Glass data, and a real-world practice dataset containing relative differential
pressure changes of a filter. For the ARMA dataset, the ARMA model performed best for
this dataset rather than other methods. For the real-world dataset, the SVM performed
slightly worse than the best Elman network, because the training phase of the SVM was
not feasible with this relatively large dataset. But by using a training set with the much
lower resolution (i.e., 10%), this method was able to anticipate the filter series well. They
concluded that the largest benefits of SVMs is the fact that a global solution exists and is
found in contrast to neural networks which have to be trained with randomly chosen
initial weight settings.
Kim (2003) has predicted the stock price index by using SVMs in his paper. He also,
has searched the effect of the upper bound C and the kernel parameter δ2 in SVM. It is
clear from experimental result that the prediction performances of SVMs are sensitive to
the value of these parameters, so finding the optimal value of the parameters is important.
In addition, they examined the feasibility of applying SVM in financial forecasting by
comparing it with back propagation neural networks and case-based reasoning. And at
the end, they concluded that SVM provides a promising alternative to stock market
prediction.
Bo et al. (2007) have proposed a recursive finite Newton algorithm for training
non-linear support vector regression (SVR-RFN). They have used the insensitive Huber
loss function (IHLF). They concluded that their method outperforms the LIBSVM 2.82.
Lu et al. (2009) have presented a two-stage forecasting model for financial time
series. At first, they used independent component (IC) analysis for generating ICs to
forecast variables. Then, ICs including noise were identified and removed. The rest of the
ICs were used as the input variables of the SVR forecasting model. They surveyed two
datasets including the Nikkei 225 opening cash index and the TAIEX closing cash index,
and compared the mentioned method with traditional SVR and random walk models by
considering prediction error and prediction accuracy as criteria. They concluded that their
method outperforms the traditional SVR and random walk models.
For improving model in order to improve the performance of the standard SVR
model, Yang et al. (2009) have presented the localised support vector regression (LSVR)
model. Their model offers a systematic and automatic scheme to adapt the margin locally
and flexibly. Hence, it can stand noise adaptively. They declared that this model
incorporates the standard SVR as a special case. And also by kernelisation, this model
can generate non-linear approximating functions, so it is able to apply to general
regression problem.
Hong et al. (2010) have forecasted Taiwanese 3G mobile phone demand by SVR with
hybrid evolutionary algorithms. They employed genetic algorithm-simulated annealing
hybrid algorithm (GA-SA) to select the suitable parameter combination for a SVR model.
Finally, they compared the results with two other models, namely the autoregressive
integrated moving average (ARIMA) model and the general regression neural networks
(GRNN) model.
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Hong (2010) has presented an application of a novel algorithm, namely chaotic ant
swarm optimisation (CAS), in a SVR-based electric load forecasting model, to improve
the forecasting performance by searching its suitable parameters combination. Finally, he
has compared the SVR model with CAS (SVRCAS) results with the other alternative
methods, namely SVRCPSO (SVR with chaotic PSO), SVRCGA (SVR with chaotic
GA), regression model, and ANN model.
Fu et al. (2010) for prediction of chaotic time series with outliers have presented
annealing robust fuzzy neural networks (ARFNNs) with SVR. They have used a
combination model that merges SVR, RBF networks and simplified fuzzy inference
system. Finally, the superiority of their method has been shown with different SVR for
training and prediction of chaotic time series with outliers.
Lu and Wang (2010) combined IC analysis and growing hierarchical self-organising
maps with SVR to forecast product demand. They used IC analysis method to detect and
remove the noise of data and further improve the performance of predicting model. Then,
they used growing hierarchical self-organising maps to classify data. After that, SVR was
applied to construct the product demand forecasting model.
Kavaklioglu (2011) have used SVR methodology to model and predict Turkey’s
electricity consumption. A grid search for the model parameters was performed to find
the best e-SVR model for each variable based on root mean square error.
Hong et al. (2011a) have used SVR to model and forecast the tourism demands with
chaotic genetic algorithm (CGA), namely SVRCGA. The proposed CGA based on the
chaos optimisation algorithm is used to overcome premature local optimum in
determining three parameters of a SVR model.
In another research, Hong et al. (2011b) used the combination of SVR with
continuous ant colony optimisation algorithms (SVRCACO) to model and forecast inter-
urban traffic flow. They compared the results with the seasonal autoregressive integrated
moving average (SARIMA) time series model.
Hong (2011) proposed a traffic flow forecasting model that combines the seasonal
SVR model with chaotic simulated annealing algorithm. Then, they compared the
results with the SARIMA time series model (SSVRCSA) to predict inter-urban traffic
flow.
Hong et al. (2011c) also used the hybrid genetic algorithm-simulated annealing
algorithm to determine the suitable parameter combination for a SVR in another research
and compared these results with SARIMA, back propagation neural network (BPNN),
Holt-Winters (HW) and seasonal Holt-Winters (SHW) models. They presented a SVR
traffic flow forecasting model.
Meiying et al. (2011) presented a Bayesian evidence framework to infer the LS-SVR
model parameters. In fact, because the traditional least squares support vector regression
(LSSVR) model (using cross validation to determine the regularisation parameter and
kernel parameter) is time-consuming, they used a Bayesian evidence framework.
Lin et al. (2011) forecasted concentrations of air pollutants by logarithm support
vector regression with immune algorithms (SVRLIA) model which takes advantage of
the structural risk minimisation of SVR models. In this investigation, three pollutants
were collected and examined to determine the feasibility of the developed SVRLIA
model.
Nagi et al. (2011) used a computational intelligence scheme based on the SOM and
SVM for the prediction of daily peak load. They used SOM as a clustering tool to cluster
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the training data into two subsets, using the Kohonen rule. Finally, they used the SVR to
fit the testing data based on the clustered subsets for predicting the daily peak load.
Rasouli et al. (2012) forecasted daily stream flow by machine learning methods –
Bayesian neural network (BNN), SVR and Gaussian process (GP) – with weather and
climate inputs. Finally, they compared the results with multiple linear regressions (MLR).
Che et al. (2012) used an adaptive fuzzy combination model based on self-organising
map and SVR for forecasting electric load.
As mentioned, for estimation of gasoline consumption in Iran railway industry, a
support vector regression is used to reach the best results in comparing with other
intelligent tools such as ANN. ANN is a powerful intelligent tool that is used in literature
for different purposes (Satapathy et al., 2012). The algorithm uses SVR and time series
model to predict Iran’s gasoline consumption. The SVR model is discussed in the next
section. The proposed formulation of model is presented in Section 4. The results of
model are discussed on a case study in Section 5. Conclusion of paper is presented in
Section 6.
4 Support vector regression
Many approaches for obtaining systems with intelligent behaviour are based on
components that learn automatically from previous experience. The development of these
learning techniques is the objective of the area of research known as machine learning.
Among the various existing algorithms, SVM for classification and SVR, after training
from a series of examples, can successfully predict the output at an unseen location
performing an operation known as induction. The machine learning process can be
divided into four categories:
• Supervised learning: it creates a function from training data which includes pairs
of input objects (typically vectors) and desired output. The output of regression
function is a continuous value, or in classification it can predict a class label of
the input object. First of all, the supervised learner is taught by a finite number of
training examples and by the use of these examples, it predicts the value of the
function for any valid input data. Nowadays supervised learning algorithms are so
popular and well-known such as: neural networks, nearest neighbour algorithm,
decision tree learning, and support vector machines.
• Unsupervised learning: as it can be guessed from its name, in this category there is
not any kind of supervisors, so there is only input data without output data (answer).
Usually the unsupervised learning problems are not mathematically well-defined and
their goal is different case by case. For instance, in data clustering, the main purpose
is to group similar data. The affinity parameter between data samples should be
subjectively predetermined and there is no objective criterion for evaluating its
validity quantitatively.
• Semi-supervised learning: it uses both unlabeled and labelled data for training,
typically a small amount of labelled data with a large amount of unlabeled data.
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• Reinforcement learning: in this category, an agent explores an environment,
perceives its current state and takes actions. The environment, in return, provides a
reward (which can be positive or negative). These algorithms attempt to find a policy
for maximising cumulative reward for the agent through the problem.
As mentioned before, SVMs form part of supervised learning create a ‘decision-maker’
system which tries to predict new values (Figure 1).
Figure 1 Different views of SVM
Source: Parrella (2007)
For conventional purposes, the group of examples that make up the SVM is called
training set, whereas the group which contains the examples used in the prediction is
called test set. SVMs can also be applied to regression problems by the introduction of an
alternative loss function. The loss function must be modified to include a distance
measure. Figure 2 illustrates four possible loss functions.
The loss function in Figure 2(a) corresponds to the conventional least squares error
criterion. The loss function in Figure 2(b) is a Laplacian loss function that is less sensitive
to outliers than the quadratic loss function. Huber proposed the loss function in
Figure 2(c) as a robust loss function that has optimal properties when the underlying
distribution of data is unknown. These three loss functions will produce no sparseness in
the support vectors. To address this issue, Vapnik proposed the loss function in
Figure 2(d) as an approximation to Huber’s loss function that enables a sparse set of
support vectors to be obtained. The ε-insensitive loss function is attractive, because the
SV solution can be sparse unlike the quadratic and Huber cost functions (that all the data
points will be support vectors).
The main purpose of SVMs is to find a function f(x) that has deviated away from all
the training data. At the same time, this function should be as flat as possible to prevent
over fitting.
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Figure 2 Loss functions, (a) quadratic (b) Laplace (c) Huber (d) ε-intensive (see online version
for colours)
(a) (b)
(c) (d)
5 Model formulation
As mentioned before, if we assume y as a single output that is a function of N input
variables x, a training dataset of length N can be presented as below:
()( )
(
)
{}
11 2 2
, , , , , , where and , 1, 2, ,
n
NN k k
Txyxy xy xR yRk N=∈∈=……
xk is n dimensional vector that demonstrates the values of each input at time step k and yk
is scalar that demonstrates the output variable at time step k. Now, the issue is finding a
model that describes this training dataset. In the standard SVR formulation a linear model
is considered as equation (1).
ˆ() ,yx wx b=〈 〉+ (1)
where the estimated output of the model is shown by ˆ,y w is the weight vector, b is the
bias term, and vector inner product is denoted by 〈.,.〉. The vector w is actually an element
of the feature space of the problem. Because of all real world problems cannot be solved
and modelled with linear formula, it allow non-linear modelling. In the SVR
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methodology kernel functions Φ(x) from input space to feature space are used as
non-linear function as presented in equation (2).
ˆ() ,Φ()
y
xwxb=+ (2)
Therefore, selecting the right non-linear maps Φ(x) and fitting for the best w is
significant issue. As mentioned before we use the RBFs because of its ability to analyse
higher-dimension data, use of just one hyper parameter to search, and fewer numerical
difficulties. To making w as flat as possible, we need to minimise the norm ( || . || ) of the
w vector for every data point i = 1, 2,…,N as presented in model (3):
()
()
2
1
Minimise 2
Subject to , Φ
,Φ
ii
ii
w
yw x bε
wx byε
−−≤
+− ≤
(3)
This model is correct only if we assume that the problem is feasible. If we want to allow
some errors, we should introduce some slack-variables *
(, )
ii
ξξ that enlarge the tolerance
of the machine as presented in model (4):
()
()
()
2*
1
*
*
1
Minimise 2
Subject to , Φ
,Φ
,0
N
ii
i
ii i
ii i
ii
wC ξξ
yw x bεξ
yw x bεξ
ξξ
=
++
−−≤+
−−≤+
≥
∑
(4)
The constant C determines the trade-off between the flatness of the function and the
amount of larger deviations of tolerance. ε is the maximum error permitted in an element,
ε and C are parameters which define the limit of maximal tolerance. Finding the right
value is very complicated and there is a vast literature on how to choose the best ones.
Generally, the most used technique is to find them by trial and error.
In fact, in the case of regression, a margin of tolerance ξ is set in approximation to the
SVM which would have already requested from the problem. However, the main idea is
always the same: to minimise error, individualising the hyper plane which maximises the
margin, keeping in mind that part of the error is tolerated. Usually this optimisation
problem is solved in its dual form, therefore constraints are carried in the cost function by
use of Lagrange multipliers and the Lagrangian (L) is created as model (5):
()
()
()
()
()
()
2*
1
1
** **
11
1
2
,Φ
,Φ
N
ii
i
N
iii i
i
NN
iii i iiii
ii
LwC ξξ
εξ yw x b
εξ yw x b ηξ η ξ
=
=
==
=+ +
−+−+ +
−++− −−+
∑
∑
∑∑
β
β
(5)
**
,,and
iii i
ηη
ββ
are the Lagrange multipliers.
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Figure 3 Graphical details of ε-insensitive loss function
Figure 3 shows how the error of SVR is calculated. Up until the threshold ξ, the error is
considered 0, after the error it becomes calculated as ‘error-epsilon’. In fact in order to
find the minimum, one needs to take all the partial derivatives of the Lagrangian
with respect to ξi, *,
i
ξ w and b; and set them to zero. These expressions along with the
complementary Karush-Kuhn-Tucker conditions lead to the final quadratic programming
(QP) model (6):
()
()
() ()
()
**
11
**
11
*
1
*
1
2
Minimise
Subject to 0
0,
NN
ii j j
ii
NN
ii iii
ii
N
ii
i
ii
εy
C
==
==
=
−−
+−− −
−=
≤≤
∑∑
∑∑
∑
ββ β β
ββ ββ
ββ
ββ
(6)
where Kij = K(xi, xj) = Φ(xi)TΦ(xj).
Kij is the kernel function based on the original non-linear maps and we do not need to
calculate w. By solving the above QP problem the optimum
β
i, *
i
β
are concluded for
i = 1, 2,…,N. Therefore, we gain a pair of
β
i, *
i
β
for all of the training data point. It is
possible that some of the
β
i, *
i
β
pair would be vanished by having zero value. Those
training data point that the
β
i, *
i
β
pair dose not vanish are a support vector. And also the
bias b is computed from this formula ˆ
(0),
ii
εyy
−
+= when this condition is satisfied for
all the support vectors. Finally, we can use the following formulation for model in the
dual space:
()
()
*
1
ˆ() ,
N
ii i
i
yx K xx b
=
=− +
∑
ββ
(7)
And the RBF is as follow:
() ()()
2
,exp 2
T
ij ij
ij i j
xx xx
KKxx σ
⎛⎞
−−
⎜⎟
==
⎝⎠
(8)
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Now we have the input parameters and the number of them (N), the regularisation
parameter (C), the maximum allowable error in the output (ε) and the kernel parameter
(σ) that determines the spread of the function.
With the use of mean absolute percentage error (MAPE) in test data, we are able to
evaluate the performance of the model and it is computed as follow:
1
ˆ
1.
Nii
ii
yy
MAPE Ny
=
−
=∑ (9)
6 Experiment
SVMs, as a set of related supervised learning methods, are a powerful learning tool for
solving classification and regression problems that analyse data and recognise patterns.
As it mentioned before, a SVR is designed for gasoline consumption in railway
transportation industry in Iran. Then, we designed and ANN model to compare the results
and consider the accuracy of SVR model. The collected gasoline dataset consists of
116 samples with three attributes. Transported freight per kilometre, transported
passengers per kilometre and the number of holidays per weeks are the inputs and weekly
gasoline consumption in Iran railway is the output. The task is to predict the consumption
of gasoline in railway transportation industry. The gasoline dataset is split into
90 training samples and 26 test samples. All dataset is scaled into the interval [–1, 1]. By
the use of SVR-RFN (Bo et al., 2007) many different values of (σ, C) has been used and
optimal values of them is assigned. Then, SVR and ANN are solved by using the
MATLAB R2010a software on a Core-i5 with 2.40 GHz. Some computational results of
the output of the models are shown in Figure 4, Figure 5 and Table 1.
In Table 1, the inputs that include number of holidays per week, transported freight
per kilometre (million tons per kilometre), transported passengers per kilometre (million
passengers per kilometre), gasoline consumption and the outputs of two models that
include predicted gasoline consumption by SVR and ANN and also their MAPE are
represented.
In Figure 4, gasoline consumption (blue line), predicted gasoline consumption by
SVR (red line) and residuals (green line) are shown. As it mentioned before, for
comparing the results of SVR, an ANN has been used to estimate and predict weekly
gasoline consumption in railway transportation industry. It has been used feed-forward
back propagation network for estimation because of its great ability to map inputs to
outputs. Figure 5 shows the results of ANN for this problem.
Figure 4 and Figure 5 show that prediction of gasoline consumption by SVR is more
realistic than prediction by ANN. Table 1 also show that the MAPE by using SVR is
12.50%, but MAPE by using ANN is 28.39%. Therefore SVR makes better prediction for
the gasoline dataset. In fact by using SVR, we can capture the trend of data and make
precise forecast and it can be very powerful tool for prediction in forecasting problems
and it may be used to estimate energy consumption in other transportation systems in all
over the world.
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Table 1 Inputs and outputs of SVR and ANN models
Number of
holidays per
weeks
Transported freight
per kilometre
Transported
passengers per
kilometre
Gasoline
consumption
Predicted gasoline
consumption by ANN
Predicted gasoline
consumption by SVR MAPE by ANN MAPE by SVR
0 376 271 328,400 292,078.96 304,525.32 11.06% 7.27%
0 416 267 194,000 234,604.2 169,226.2 20.93% 12.77%
0 448 279 279,250 234,234.9 321,640.15 16.12% 15.18%
0 419 300 198,900 155,778.48 186,409.08 21.68% 6.28%
1 433 350 202,100 171,966.89 241,165.93 14.91% 19.33%
0 427 331 546,515 562,527.8895 496,782.135 2.93% 9.10%
0 413 307 556,142 581,613.3036 576,886.0966 4.58% 3.73%
1 390 320 578,100 547,980.99 547,345.08 5.21% 5.32%
0 425 307 805,648 821,760.96 822,808.3024 2.00% 2.13%
1 419 308 2,696,993 3,144,349.129 2,506,854.994 16.59% 7.05%
1 401 312 3,054,292 3,289,122.222 2,838,658.985 7.69% 7.06%
0 433 308 3,352,113 3,296,133.043 4,438,264.789 1.67% 32.40%
0 411 306 3,717,584 5,015,568.606 4,293,615.35 34.91% 15.49%
0 387 315 4,685,795 5,056,807.503 2,890,072.882 7.92% 38.32%
1 397 333 4,996,361 4,736,550.228 3,711,055.39 5.20% 25.72%
1 368 330 4,732,666 3,631,024.508 4,914,760.684 23.28% 3.85%
0 379 305 4,982,343 5,173,157.322 5,802,922.053 3.83% 16.47%
0 395 291 5,728,337 3,803,883.842 6,452,249.912 33.60% 12.64%
2 414 275 6,209,073 5,855,155.839 6,962,331.407 5.70% 12.13%
0 400 263 6,254,518 9,623,201.464 7,011,265.358 53.86% 12.10%
0 444 262 6,001,143 3,114,193.841 4,507,631.234 48.11% 24.89%
0 401 270 5,951,384 8,583,522.307 5,762,385.827 44.23% 3.18%
0 396 297 5,834,499 8,242,885.569 6,840,850.804 41.28% 17.25%
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Table 1 Inputs and outputs of SVR and ANN models (continued)
Number of
holidays per
weeks
Transported freight
per kilometre
Transported
passengers per
kilometre
Gasoline
consumption
Predicted gasoline
consumption by ANN
Predicted gasoline
consumption by SVR MAPE by ANN MAPE by SVR
0 385 302 6,200,574 8,580,310.508 6,279,933.685 38.38% 1.28%
1 403 318 6,157,591 3,704,434.579 6,436,197.135 39.84% 4.52%
2 401 330 6,097,827 4,862,456.262 6,406,228.495 20.26% 5.06%
0 405 307 6,142,619 6,726,167.805 5,516,397.844 9.50% 10.19%
1 414 319 6,299,457 9,242,062.508 6,837,498.909 46.71% 8.54%
0 381 310 6,142,215 6,480,036.825 5,879,436.016 5.50% 4.28%
0 438 315 6,181,518 8,629,623.818 6,274,417.29 39.60% 1.50%
5 420 325 6,511,955 8,977,617.544 7,094,519.527 37.86% 8.95%
1 419 366 6,346,846 5,940,647.856 5,390,154.325 6.40% 15.07%
0 438 314 6,310,899 3,692,150.926 6,606,195.918 41.50% 4.68%
0 445 288 6,404,782 3,652,910.701 7,882,009.461 42.97% 23.06%
0 468 302 6,298,429 6,720,423.743 6,075,270.692 6.70% 3.54%
0 434 311 6,482,897 3,627,891.96 8,000,495.561 44.04% 23.41%
0 479 306 6,397,869 9,650,584.495 6,706,823.363 50.84% 4.83%
0 413 317 6,317,499 7,414,766.097 8,935,821.86 17.37% 41.45%
1 393 315 5,859,065 8,386,246.754 7,694,092.043 43.13% 31.32%
0 395 305 5,963,647 8,176,093.991 7,409,198.102 37.10% 24.24%
0 437 326 6,148,955 3,331,128.28 5,761,189.615 45.83% 6.31%
1 403 327 5,929,462 4,368,388.375 4,956,241.306 26.33% 16.41%
0 428 343 6,049,008 3,994,303.747 5,888,058.058 33.97% 2.66%
0 413 336 5,990,294 7,394,460.829 5,107,142.256 23.44% 14.74%
1 378 344 5,921,860 3,336,232.038 5,872,180.917 43.66% 0.84%
0 433 361 5,906,986 4,050,681.671 5,717,932.902 31.43% 3.20%
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Table 1 Inputs and outputs of SVR and ANN models (continued)
Number of
holidays per
weeks
Transported freight
per kilometre
Transported
passengers per
kilometre
Gasoline
consumption
Predicted gasoline
consumption by ANN
Predicted gasoline
consumption by SVR MAPE by ANN MAPE by SVR
1 398 364 5,945,310 3,625,495.053 6,905,240.864 39.02% 16.15%
0 412 375 5,986,946 5,024,058.716 6,705,066.118 16.08% 11.99%
1 345 380 6,076,471 8,251,843.6 7,755,954.883 35.80% 27.64%
0 356 378 6,142,379 8,640,786.432 6,017,900.869 40.67% 2.03%
0 413 343 6,114,930 4,280,917.903 7,772,767.523 29.99% 27.11%
0 399 252 5,963,808 8,106,527.001 3,128,881.997 35.93% 47.54%
0 431 262 6,093,792 8,344,427.335 3,941,732.107 36.93% 35.32%
1 414 263 6,100,340 8,808,534.256 5,990,454.94 44.39% 1.80%
0 419 324 6,055,007 5,533,810.448 8,104,757.883 8.61% 33.85%
0 400 388 6,202,582 4,505,285.766 6,756,987.63 27.36% 8.94%
0 439 394 6,158,782 6,454,403.536 5,469,179.873 4.80% 11.20%
0 399 349 6,290,364 4,211,613.709 5,230,062.337 33.05% 16.86%
1 423 348 6,219,366 4,810,931.646 5,807,117.736 22.65% 6.63%
0 424 340 6,079,518 8,569,450.698 6,054,193.501 40.96% 0.42%
0 462 350 6,203,760 3,340,951.444 6,351,238.539 46.15% 2.38%
0 441 346 6,382,580 9,364,872.795 6,460,321.728 46.73% 1.22%
0 484 345 6,338,768 2,434,807.296 6,401,433.637 61.59% 0.99%
0 423 338 6,363,702 6,726,433.014 5,897,746.873 5.70% 7.32%
1 433 358 6,610,467 9,917,773.002 8,921,058.064 50.03% 34.95%
1 416 352 6,396,451 9,180,052.326 5,394,087.565 43.52% 15.67%
0 416 341 6,359,159 9,385,459.121 6,928,511.951 47.59% 8.95%
0 434 323 6,311,278 5,942,249.276 5,971,623.128 5.85% 5.38%
2 362 310 6,252,551 7,582,758.069 6,834,287.775 21.27% 9.30%
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. Azadeh et al.
Table 1 Inputs and outputs of SVR and ANN models (continued)
Number of
holidays per
weeks
Transported freight
per kilometre
Transported
passengers per
kilometre
Gasoline
consumption
Predicted gasoline
consumption by ANN
Predicted gasoline
consumption by SVR MAPE by ANN MAPE by SVR
0 457 336 6,213,050 3,993,737.505 5,270,110.078 35.72% 15.18%
0 458 300 6,242,344 3,485,693.023 5,681,309.944 44.16% 8.99%
0 408 286 6,067,530 5,697,410.67 8,588,452.831 6.10% 41.55%
0 418 289 5,797,463 8,188,561.551 5,351,058.349 41.24% 7.70%
0 399 306 6,240,853 8,656,943.825 5,263,329.229 38.71% 15.66%
1 381 334 6,240,232 3,305,624.805 5,212,054.276 47.03% 16.48%
1 384 349 6,260,662 4,035,611.75 6,059,268.66 35.54% 3.22%
0 421 360 6,233,075 3,268,201.677 5,288,712.347 47.57% 15.15%
0 422 367 6,205,547 9,749,500.26 7,201,625.112 57.11% 16.05%
1 391 355 6,372,346 8,112,620.892 7,012,866.436 27.31% 10.05%
0 439 331 6,334,043 6,796,428.139 5,856,110.623 7.30% 7.55%
0 430 324 6,295,672 5,221,579.94 5,368,095.269 17.06% 14.73%
0 433 328 6,207,819 6,536,833.407 6,303,765.487 5.30% 1.55%
5 430 322 6,411,659 9,680,550.14 5,977,764.248 50.98% 6.77%
0 397 369 6,336,286 9,216,332.892 5,510,648.121 45.45% 13.03%
1 382 338 6,070,806 9,036,281.81 5,602,990.97 48.85% 7.71%
0 372 291 6,074,187 8,075,280.774 6,793,730.599 32.94% 11.85%
0 445 302 6,280,505 7,789,183.865 6,171,803.747 24.02% 1.73%
0 441 331 6,240,607 9,763,136.939 5,528,435.884 56.45% 11.41%
0 453 330 6,295,008 5,931,057.433 7,443,920.29 5.78% 18.25%
1 441 348 6,174,216 5,791,414.608 7,969,506.501 6.20% 29.08%
0 418 323 6,344,598 6,052,746.492 3,871,135.612 4.60% 38.99%
0 398 315 6,078,317 8,033,873.635 4,195,378.975 32.17% 30.98%
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Table 1 Inputs and outputs of SVR and ANN models (continued)
Number of
holidays per
weeks
Transported freight
per kilometre
Transported
passengers per
kilometre
Gasoline
consumption
Predicted gasoline
consumption by ANN
Predicted gasoline
consumption by SVR MAPE by ANN MAPE by SVR
0 422 350 5,928,236 8,489,822.168 6,753,083.16 43.21% 13.91%
2 398 352 6,038,828 7,527,986.158 6,191,221.639 24.66% 2.52%
1 381 355 5,959,803 7,960,056.649 5,635,433.606 33.56% 5.44%
0 371 367 6,080,230 8,000,048.618 6,180,752.883 31.57% 1.65%
1 357 377 5,878,211 5,355,050.221 5,763,202.826 8.90% 1.96%
0 384 375 6,027,713 8,466,447.265 6,422,361.063 40.46% 6.55%
0 417 378 6,301,421 9,128,384.897 5,826,282.525 44.86% 7.54%
1 407 380 6,504,907 3,601,960.43 6,412,496.082 44.63% 1.42%
0 373 381 6,367,646 6,042,896.054 7,700,758.317 5.10% 20.94%
0 400 304 6,285,312 4,206,098.923 5,713,995.845 33.08% 9.09%
0 422 229 6,574,721 4,510,466.551 7,111,300.126 31.40% 8.16%
0 448 278 6,531,711 6,707,505.238 8,606,243.424 2.69% 31.76%
1 443 269 6,448,218 6,783,525.336 6,466,093.004 5.20% 0.28%
1 454 364 6,752,270 7,110,140.31 6,534,956.278 5.30% 3.22%
0 404 397 6,638,044 3,220,496.051 5,467,887.859 51.48% 17.63%
0 419 401 6,571,895 6,256,444.04 7,464,785.501 4.80% 13.59%
0 448 481 6,512,153 3,519,062.703 8,237,717.67 45.96% 26.50%
1 391 479 6,312,874 3,540,688.46 6,443,102.132 43.91% 2.06%
0 361 353 6,146,221 7,053,963.042 5,815,206.101 14.77% 5.39%
0 373 372 6,076,737 8,212,523.181 5,832,920.259 35.15% 4.01%
0 366 362 6,090,438 8,663,461.755 5,988,165.453 42.25% 1.68%
0 400 380 5,995,286 8,374,101.596 6,195,704.112 39.68% 3.34%
0 395 385 5,806,401 6,090,914.649 6,342,805.598 4.90% 9.24%
1 332 310 5,739,097 7,662,073.708 5,426,696.142 33.51% 5.44%
234
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. Azadeh et al.
Figure 5 The predicted line by ANN for dataset (see online version for colours)
Figure 4 The SVR by RBF kernel function for dataset (see online version for colours)
7 Conclusions
This study presented a SVR algorithm to estimate and predict weekly gasoline
consumption in railway transportation industry in Iran. Furthermore, it considered the
effect of number of holidays per weeks and amount of transported loads and number of
transported passengers in gasoline consumption prediction. This is the first study that
integrates conventional time series and SVR for forecasting and modelling gasoline
consumption in railway industry in Iran. Weekly gasoline consumption in railway
transportation industry from August 2009 to December 2011 was considered. By
comparing SVR’s results with other intelligent tools such as ANN, it was revealed that
SVR makes better prediction results for dataset. Therefore, managers could use SVR for
accurate prediction of gasoline consumption for making strategic decisions. Using
meta-heuristic approaches such as PSO, GA, and etc., in determining three parameters of
a SVR model can be an extension to this study. In this paper, we used transported
passengers per kilometre, transported tons per kilometre, and holidays in weeks as inputs
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of our model. We can use the other factors and compare the results to determine the most
important and relevant factors for prediction of gasoline consumption as future research.
Acknowledgements
The authors are grateful for the valuable comments and suggestion from the respected
reviewers. Their valuable comments and suggestions have enhanced the strength and
significance of our paper. This study was supported by a grant from University of Tehran
(Grant No. 8106013/1/14). The authors are grateful for the support provided by the
College of Engineering, University of Tehran, Iran.
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